2007B AMC 12 — Official Competition Problems (February 2024)
📅 2024 A 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
几何·面积
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted? (A)\ 678 (B)\ 768 (C)\ 786 (D)\ 867 (E)\ 876
💡 解题思路
There are four walls in each bedroom (she can't paint floors or ceilings). Therefore, we calculate the number of square feet of walls there is in one bedroom: \[2\cdot(12\cdot8+10\cdot8)-60=2\cdot176-
2
第 2 题
统计
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
💡 解题思路
The trip was $240$ miles long and took $\dfrac{120}{30}+\dfrac{120}{20}=4+6=10$ gallons. Therefore, the average mileage was $\dfrac{240}{10}= \boxed{\textbf{(B) }24}$
3
第 3 题
几何·面积
The point O is the center of the circle circumscribed about triangle ABC , with \angle BOC = 120^{\circ} and \angle AOB = 140^{\circ} , as shown. What is the degree measure of \angle ABC ? \mathrm {(A)} 35 \mathrm {(B)} 40 \mathrm {(C)} 45 \mathrm {(D)} 50 \mathrm {(E)} 60
💡 解题思路
Since triangles $ABO$ and $BOC$ are isosceles, $\angle ABO=20^o$ and $\angle OBC=30^o$ . Therefore, $\angle ABC=50^o$ , or $\mathrm{(D)}$ .
4
第 4 题
统计
At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?
💡 解题思路
$3$ bananas cost as much as $2$ apples, so $18$ bananas cost as much as $12$ apples. Since $6$ apples cost as much as $4$ oranges, $12$ apples cost as much as $8$ oranges, giving $(B)$ .
5
第 5 题
方程
The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points? \mathrm {(A)}\ 13 \mathrm {(B)}\ 14 \mathrm {(C)}\ 15 \mathrm {(D)}\ 16 \mathrm {(E)} 17
💡 解题思路
She must get at least $100 - 4.5 = 95.5$ points, and that can only be possible by answering at least $\lceil \frac{95.5}{6}\rceil = 16 \Rightarrow \mathrm {(D)}$ questions correctly.
6
第 6 题
几何·面积
Triangle ABC has side lengths AB = 5 , BC = 6 , and AC = 7 . Two bugs start simultaneously from A and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point D . What is BD ? \mathrm {(A)}\ 1 \mathrm {(B)}\ 2 \mathrm {(C)}\ 3 \mathrm {(D)}\ 4 \mathrm {(E)}\ 5
💡 解题思路
One bug goes to $B$ . The path that he takes is $\dfrac{5+6+7}{2}=9$ units long. The length of $BD$ is $9-AB=9-5=4 \Rightarrow \mathrm {(D)}$
7
第 7 题
几何·角度
All sides of the convex pentagon ABCDE are of equal length, and \angle A = \angle B = 90^{\circ} . What is the degree measure of \angle E ? \mathrm {(A)}\ 90 \mathrm {(B)}\ 108 \mathrm {(C)}\ 120 \mathrm {(D)}\ 144 \mathrm {(E)}\ 150
💡 解题思路
Since $A$ and $B$ are right angles , and $AE$ equals $BC$ , and $AECB$ is a square . Since the length of $ED$ and $CD$ are also 5, triangle $CDE$ is equilateral . Angle $E$ is therefore $90+60=150 \Ri
8
第 8 题
规律与数列
Tom's age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of their ages then. What is T/N ?
💡 解题思路
Tom's age $N$ years ago was $T-N$ . The sum of the ages of his three children $N$ years ago was $T-3N,$ since there are three children. If his age $N$ years ago was twice the sum of the children's age
9
第 9 题
函数
A function f has the property that f(3x-1)=x^2+x+1 for all real numbers x . What is f(5) ? (A)\ 7 (B)\ 13 (C)\ 31 (D)\ 111 (E)\ 211
💡 解题思路
$3x-1 =5 \implies x= 2$
10
第 10 题
应用题
Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40\% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30\% of the group are girls. How many girls were initially in the group?
💡 解题思路
If we let $p$ be the number of people initially in the group, then $0.4p$ is the number of girls. If two girls leave and two boys arrive, the number of people in the group is still $p$ , but the numbe
11
第 11 题
几何·角度
The angles of quadrilateral ABCD satisfy \angle A=2 \angle B=3 \angle C=4 \angle D. What is the degree measure of \angle A, rounded to the nearest whole number?
💡 解题思路
The sum of the interior angles of any quadrilateral is $360^\circ.$ \begin{align*} 360 &= \angle A + \angle B + \angle C + \angle D\\ &= \angle A + \frac{1}{2}A + \frac{1}{3}A + \frac{1}{4}A\\ &= \fra
12
第 12 题
统计
A teacher gave a test to a class in which 10\% of the students are juniors and 90\% are seniors. The average score on the test was 84. The juniors all received the same score, and the average score of the seniors was 83. What score did each of the juniors receive on the test?
💡 解题思路
We can assume there are $10$ people in the class. Then there will be $1$ junior and $9$ seniors. The sum of everyone's scores is $10 \cdot 84 = 840$ . Since the average score of the seniors was $83$ ,
13
第 13 题
概率
A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? (A)\ \frac{1}{63} (B)\ \frac{1}{21} (C)\ \frac{1}{10} (D)\ \frac{1}{7} (E)\ \frac{1}{3}
💡 解题思路
The traffic light runs through a $63$ second cycle.
14
第 14 题
几何·面积
Point P is inside equilateral \triangle ABC . Points Q , R , and S are the feet of the perpendiculars from P to \overline{AB} , \overline{BC} , and \overline{CA} , respectively. Given that PQ=1 , PR=2 , and PS=3 , what is AB ? (A)\ 4 (B)\ 3√(3) (C)\ 6 (D)\ 4√(3) (E)\ 9
💡 解题思路
Drawing $\overline{PA}$ , $\overline{PB}$ , and $\overline{PC}$ , $\triangle ABC$ is split into three smaller triangles. The altitudes of these triangles are given in the problem as $PQ$ , $PR$ , and
15
第 15 题
规律与数列
The geometric series a+ar+ar^2\ldots has a sum of 7 , and the terms involving odd powers of r have a sum of 3 . What is a+r ?
💡 解题思路
The sum of an infinite geometric series is given by $\frac{a}{1-r}$ where $a$ is the first term and $r$ is the common ratio.
16
第 16 题
几何·角度
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible? (A)\ 15 (B)\ 18 (C)\ 27 (D)\ 54 (E)\ 81 https://youtu.be/0W3VmFp55cM?t=5067 ~ pi_is_3.14
💡 解题思路
A tetrahedron has 4 sides. We can ignore the rotation part of the question completely and focus on the colors. The ratio of the number of faces with each color must be one of the following:
17
第 17 题
统计
If a is a nonzero integer and b is a positive number such that ab^2=\log_{10}b , what is the median of the set \{0,1,a,b,1/b\} ? (A)\ 0 (B)\ 1 (C)\ a (D)\ b (E)\ \frac{1}{b}
💡 解题思路
Note that if $a$ is positive, then, the equation will have no solutions for $b$ . This becomes more obvious by noting that at $a=1$ , $ab^2 > \log_{10} b$ . The LHS quadratic function will increase fa
18
第 18 题
几何·面积
Let a , b , and c be digits with a\ne 0 . The three-digit integer abc lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer acb lies two thirds of the way between the same two squares. What is a+b+c ? (A)\ 10 (B)\ 13 (C)\ 16 (D)\ 18 (E)\ 21
💡 解题思路
The difference between $acb$ and $abc$ is given by
19
第 19 题
几何·角度
Rhombus ABCD , with side length 6 , is rolled to form a cylinder of volume 6 by taping \overline{AB} to \overline{DC} . What is \sin(\angle ABC) ? (A)\ \frac{π}{9} (B)\ \frac{1}{2} (C)\ \frac{π}{6} (D)\ \frac{π}{4} (E)\ \frac{√(3)}{2}
The parallelogram bounded by the lines y=ax+c , y=ax+d , y=bx+c , and y=bx+d has area 18 . The parallelogram bounded by the lines y=ax+c , y=ax-d , y=bx+c , and y=bx-d has area 72 . Given that a , b , c , and d are positive integers, what is the smallest possible value of a+b+c+d ? \mathrm {(A)} 13 \mathrm {(B)} 14 \mathrm {(C)} 15 \mathrm {(D)} 16 \mathrm {(E)} 17
💡 解题思路
Plotting the parallelogram on the coordinate plane, the 4 corners are at $(0,c),(0,d),\left(\frac{d-c}{a-b},\frac{ad-bc}{a-b}\right),\left(\frac{c-d}{a-b},\frac{bc-ad}{a-b}\right)$ . Because $72= 4\cd
21
第 21 题
整数运算
The first 2007 positive integers are each written in base 3 . How many of these base- 3 representations are palindromes? (A palindrome is a number that reads the same forward and backward.) \mathrm {(A)}\ 100 \mathrm {(B)}\ 101 \mathrm {(C)}\ 102 \mathrm {(D)}\ 103 \mathrm {(E)}\ 104
💡 解题思路
$2007_{10} = 2202100_{3}$
22
第 22 题
几何·面积
Two particles move along the edges of equilateral \triangle ABC in the direction \[A\Rightarrow B\Rightarrow C\Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at A , and the other starts at the midpoint of \overline{BC} . The midpoint of the line segment joining the two particles traces out a path that encloses a region R . What is the ratio of the area of R to the area of \triangle ABC ? \mathrm {(A)} \frac{1}{16} \mathrm {(B)} \frac{1}{12} \mathrm {(C)} \frac{1}{9} \mathrm {(D)} \frac{1}{6} \mathrm {(E)} \frac{1}{4}
💡 解题思路
First, notice that each of the midpoints of $AB$ , $BC$ , and $CA$ are on the locus. Suppose after some time the particles have each been displaced by a short distance $x$ , to new positions $A'$ and
23
第 23 题
几何·面积
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to 3 times their perimeters? \mathrm {(A)} 6 \mathrm {(B)} 7 \mathrm {(C)} 8 \mathrm {(D)} 10 \mathrm {(E)} 12
💡 解题思路
Let $a$ and $b$ be the two legs of the triangle.
24
第 24 题
数论
Also refer to the 2007 AMC 10B #25 (same problem) How many pairs of positive integers (a,b) are there such that gcd(a,b)=1 and \frac{a}{b} + \frac{14b}{9a} is an integer? \mathrm {(A)}\ 4 \mathrm {(B)}\ 6 \mathrm {(C)}\ 9 \mathrm {(D)}\ 12 \mathrm {(E)}\ infinitely many
💡 解题思路
Combining the fraction, $\frac{9a^2 + 14b^2}{9ab}$ must be an integer.
25
第 25 题
几何·面积
Points A,B,C,D and E are located in 3-dimensional space with AB=BC=CD=DE=EA=2 and \angle ABC=\angle CDE=\angle DEA=90^o . The plane of \triangle ABC is parallel to \overline{DE} . What is the area of \triangle BDE ? \mathrm {(A)} √(2) \mathrm {(B)} √(3) \mathrm {(C)} 2 \mathrm {(D)} √(5) \mathrm {(E)} √(6)